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As @Ilya says, the limit is zero due to the asymptotic size of maximum cliques/independent sets. However, there is another way to see this using endomorphisms (homomorphisms from a graph to itself). …

It doesn't imply that the categorical product of critical graphs is critical, and this is not true. For instance, the complete graph on three vertices, $K_3$, is 3-critical, but $K_3 \times K_3$ is no …

Things are even worse than you imagine. Tony's answer shows that the homomorphism order of line graphs can be partitioned into intervals whose endpoints are the complete graphs. I made use of this fa …

Apparently, Schrijver defined a function $\vartheta'$ in the paper A Comparison of the Delsart and Lovasz Bounds which (I am pretty sure) is equivalent to vector chromatic number of the complement. Fu …

For $\alpha \ge 2$, an $\alpha$-vector coloring of a graph $X$ is an assignment of unit vectors to the vertices of $X$ such that vectors assigned to adjacent vertices have inner product less than or e …

I recently constructed the graph shown below in the process of investigating some problems regarding line graphs and homomorphisms, and then happened to see it on wikipedia. I was wondering if anyone …